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arxiv: 2605.06882 · v1 · submitted 2026-05-07 · 💻 cs.AI

Recognition: 2 theorem links

· Lean Theorem

How Well Do LLMs Perform on the Simplest Long-Chain Reasoning Tasks: An Empirical Study on the Equivalence Class Problem

Chun Zheng , Lianlong Wu , Bingqian Li , Lvting Liu , Yi Zhou
Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:26 UTC · model grok-4.3

classification 💻 cs.AI
keywords large language modelsreasoning tasksequivalence class problemlong-chain reasoningempirical evaluationtransitive inference
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The pith

Non-reasoning large language models fail to determine variable equivalence from given relations, while reasoning models improve but still fall short.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests how well LLMs handle the Equivalence Class Problem, a task of checking if two variables are equivalent based on a list of equality statements that form chains. It shows that models without explicit reasoning capabilities perform at chance levels or worse on these tasks. Reasoning-enhanced models achieve higher accuracy but never reach perfect performance across all tested cases. The results highlight that even this basic transitive reasoning remains challenging for current LLMs.

Core claim

The authors establish that the Equivalence Class Problem serves as a minimal test for long-chain reasoning, where non-reasoning LLMs consistently fail to track equalities correctly across multiple relations, reasoning LLMs show marked improvement yet still err on a substantial fraction of instances, and the peak difficulty for reasoning models occurs when the relation graph has its largest diameter.

What carries the argument

The Equivalence Class Problem (ECP), which requires inferring whether two variables belong to the same class by chaining multiple equivalence statements.

Load-bearing premise

The randomly generated equivalence relations with different connectivity probabilities represent a fair test of long-chain reasoning ability without being confounded by other model limitations.

What would settle it

A demonstration that some LLM solves every ECP instance correctly for numbers of variables up to 20 and all connectivity probabilities would falsify the claim that LLMs struggle to completely solve this problem.

Figures

Figures reproduced from arXiv: 2605.06882 by Bingqian Li, Chun Zheng, Lianlong Wu, Lvting Liu, Yi Zhou.

Figure 1
Figure 1. Figure 1: Scaling Limit. Maximum error rate vs. problem size (n). The left panel reports non-reasoning models, whose peak error rises sharply as the variable set grows. The right panel reports reasoning models, which substantially reduce the error scale but still expose nonzero failures under larger instances. Each point is the maximum error observed over the swept probability range for that problem size. increasing… view at source ↗
Figure 2
Figure 2. Figure 2: Error Rate vs. Probability Analysis (n = 89 and n = 144). The x-axis is the edge probability p used to sample equivalence relations; the y-axis is query error rate. The two subfigures compare DeepSeek-V3.2 and DeepSeek-R1-250528 at n = 89 and n = 144. Vertical reference markers indicate the sparse-chain regime around 1/n and the connectivity threshold around ln n/(n−1), where the model failure modes concen… view at source ↗
Figure 3
Figure 3. Figure 3: Inference Depth Analysis (n = 89 and n = 144). The x-axis is the shortest-path inference depth between the queried variables, and the y-axis is the corresponding average error rate. DeepSeek-V3.2 collapses immediately beyond single-hop inference (Depth ≥ 2), whereas DeepSeek-R1-250528 keeps errors low but still shows depth-dependent accumulation. error rate of ∼4.4% at Depth=1, suggesting robust per￾forman… view at source ↗
Figure 4
Figure 4. Figure 4: Contextual Robustness of DeepSeek-R1 (n = 144 and n = 200). Comparison of error rates under different batching strategies: 100 × 1 (1 query per graph), 20 × 5, and 10 × 10. Left (n = 144): The 10 × 10 setting (Orange) shows higher error variance compared to the stable 100 × 1 setting (Blue). Right (n = 200): As complexity increases, the gap widens. Notably, even the cleanest 100 × 1 setting fails to solve … view at source ↗
Figure 5
Figure 5. Figure 5: Ablation Studies (n = 144). (a) Explicitly stating logical rules (P1) yields performance nearly identical to implicit prompts (P3). (b) Framing the problem as a Graph Theory task (P4) surprisingly outperforms the abstract Logic framing (P1). (c) Few-shot examples (P5) demonstrate comparable performance to Zero-shot (P1), indicating that ICL offers limited benefits for global logical reasoning. (d) Higher t… view at source ↗
read the original abstract

Large Language Models (LLMs) have achieved great improvements in recent years. Nevertheless, it still remains unclear how good LLMs are for reasoning tasks, especially for long-chain ones. In this paper, we evaluate LLMs' performance on the simplest yet long-chain reasoning task, namely the Equivalence Class Problem (ECP), i.e., determining whether two variables are equal given a set of randomly generated equivalence relations. We consider both reasoning and non-reasoning representative LLMs over a large variety of problem instances, ranging over different numbers of variables, connectivity probabilities, prompts, and other factors. The experimental results show that non-reasoning LLMs fail ECP, while reasoning models are significantly better but still struggle to completely solve this problem. Interestingly, considering various connectivity probabilities with a fixed number of variables, we observe that, for non-reasoning models, the hardest problem instances coincide with the phase transition point of ln n/(n-1), suggesting the chaos of the problem; in contrast, for reasoning models, the hardest ones coincide with the biggest diameter, suggesting the reasoning difficulty of the problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper empirically evaluates LLMs on the Equivalence Class Problem (ECP): given randomly generated equivalence relations over n variables, determine whether two specified variables are equivalent. It tests representative reasoning and non-reasoning models across varying n, connectivity probabilities p, and prompts. Main results: non-reasoning models largely fail ECP while reasoning models perform substantially better yet remain imperfect; for non-reasoning models the accuracy minimum occurs at the phase-transition p = ln n/(n-1), interpreted as evidence of structural chaos, whereas for reasoning models the minimum occurs at maximum diameter, interpreted as evidence of reasoning difficulty.

Significance. If the reported performance gaps and structural correlations survive controls for prompt length and output parsing, the work supplies concrete, falsifiable measurements of LLM limitations on a minimal long-chain reasoning task that maps directly to union-find connectivity. The distinction between model classes and the alignment of difficulty peaks with graph-theoretic quantities (phase transition vs. diameter) could guide targeted improvements in reasoning architectures and evaluation benchmarks.

major comments (2)
  1. [results on connectivity probabilities] The interpretation that non-reasoning models' accuracy minimum at p = ln n/(n-1) reflects 'chaos' of the equivalence relation is undermined by the absence of length-matched controls. Because expected statement count (and therefore prompt token length) scales linearly with p, the observed dip at the phase-transition regime may arise from attention dilution or parsing load rather than graph structure per se. This directly affects the central claim about structural difficulty (see the paragraph on varying connectivity probabilities with fixed n).
  2. [experimental setup] The abstract and experimental description report clear directional trends across model classes but provide no information on the number of independent trials per condition, statistical significance tests, exact prompt templates, or procedures for parsing model outputs. Without these, the quantitative performance gaps and the claimed alignment with phase transition or diameter cannot be assessed for robustness.
minor comments (2)
  1. Add a table that reports accuracy (with standard error) for each model class across all tested (n, p) pairs to make the trends easier to inspect.
  2. [problem generation] Clarify in the methods whether the two query variables are chosen uniformly or with any bias relative to the generated equivalence classes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our empirical study of LLMs on the Equivalence Class Problem. The comments identify important gaps in controls and reporting that we will address. Below we respond point by point to the major comments.

read point-by-point responses

  1. Referee: [results on connectivity probabilities] The interpretation that non-reasoning models' accuracy minimum at p = ln n/(n-1) reflects 'chaos' of the equivalence relation is undermined by the absence of length-matched controls. Because expected statement count (and therefore prompt token length) scales linearly with p, the observed dip at the phase-transition regime may arise from attention dilution or parsing load rather than graph structure per se. This directly affects the central claim about structural difficulty (see the paragraph on varying connectivity probabilities with fixed n).

    Authors: We agree that the linear scaling of statement count (and thus prompt length) with p introduces a plausible confound that could affect attention or parsing rather than reflecting graph structure alone. The original manuscript did not include length-matched controls. We will revise by adding a dedicated control experiment that holds prompt length approximately constant across p values (via fixed statement counts or neutral padding) while varying connectivity. Results from this control will be reported alongside the original curves; if the phase-transition dip persists, we will retain the structural-chaos interpretation with appropriate caveats; if it does not, we will qualify the claim accordingly. revision: yes

  2. Referee: [experimental setup] The abstract and experimental description report clear directional trends across model classes but provide no information on the number of independent trials per condition, statistical significance tests, exact prompt templates, or procedures for parsing model outputs. Without these, the quantitative performance gaps and the claimed alignment with phase transition or diameter cannot be assessed for robustness.

    Authors: We acknowledge that these methodological details were omitted. In the revised manuscript we will add: (i) the exact number of independent trials per (n, p, model) condition, (ii) the statistical tests performed (including p-values for key comparisons), (iii) the full prompt templates in an appendix, and (iv) a precise description of the output-parsing procedure (keyword extraction supplemented by manual review of edge cases). These additions will enable readers to evaluate the robustness of the reported trends and alignments. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical measurements on generated instances

full rationale

The paper reports direct experimental results from prompting LLMs on freshly sampled equivalence-class instances across varying n and connectivity probability p. No derivation, equation, or central claim reduces to a fitted parameter, self-citation loop, or ansatz imported from the authors' prior work. The noted coincidence with the known random-graph connectivity threshold ln n/(n-1) is an external reference, not a self-derived result. The study is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Empirical benchmarking paper. No mathematical derivation, no fitted parameters inside a model, no new postulated entities. The only background assumptions are that ECP instances require genuine long-chain reasoning and that random-graph connectivity controls difficulty in a meaningful way.

axioms (1)
  • domain assumption The equivalence class problem is the simplest long-chain reasoning task
    Stated in the title and abstract as the premise for choosing this benchmark.

pith-pipeline@v0.9.0 · 5512 in / 1312 out tokens · 27479 ms · 2026-05-11T01:26:13.623423+00:00 · methodology

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Reference graph

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